$\text{K}$-Lorentzian and $\text{K}$-CLC Polynomials in Stability Analysis

We study the class of $\text{K}$-Lorentzian polynomials, a generalization of the distinguished class of Lorentzian polynomials. As shown in \cite{GPlorentzian}, the set of $\text{K}$-Lorentzian polynomials is equivalent to the set of $\text{K}$-completely log-concave (aka $\text{K}$-CLC) forms. Throughout this paper, we interchangeably use the terms $\text{K}$-Lorentzian polynomials for the homogeneous setting and $\text{K}$-CLC polynomials for the non-homogeneous setting. By introducing an alternative definition of $\text{K}$-CLC polynomials through univariate restrictions, we establish that any strictly $\text{K}$-CLC polynomial of degree $d\le 4$ is Hurwitz-stable polynomial over $\text{K}$. Additionally, we characterize the conditions under which a strictly $\text{K}$-CLC of degree $d\ge 5$ is Hurwitz-stable over $\text{K}$. Furthermore, we associate the largest possible proper cone, denoted by $\text{K}(f,v)$, with a given $\text{K}$-Lorentzian polynomial $f$ in the direction $v\in \text{inter}\text{K}$. Finally, we investigate applications of $\text{K}$-CLC polynomials in the stability analysis of evolution variational inequalities (EVI) dynamical systems governed by differential equations and inequality constraints.